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Theory of Probability and its Applications

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1958

Volume 3, Issue 4, pp. 335-440


Limit Theorems for Markov Chains with a Finite Number of States

L. D. Meshalkin

Theory Probab. Appl. 3, pp. 335-357 (23 pages) | Cited 2 times

Online Publication Date: July 17, 2006

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Consider the scheme of trial sequences\[ \begin{gathered} \nu _{11} \hfill \\ \nu _{21} ,\nu _{22} \hfill \\ \cdots \hfill \\ \nu _{n1} ,\nu _{n2} , \cdots ,\nu _{nn} \hfill \\ \cdots \cdots \cdots \hfill \\ \end{gathered} \]The sequence $\nu _{nk} $, $k = 1, \cdots ,n$, is a uniform Markov chain with a finite number of states $E_1 , \cdots ,E_s $ and a given matrix of transition probabilities\[ P = P(n) = \left\| {p_{uv} (n)} \right\|_{u,v = 1}^s . \]
Let $\mu = \mu (n)$ denote the number of passages up in the $n$-th sequence of trials of the system through $E_1 $ on condition that the system is in state $E_1 $ at the initial (or zero-th) time. We consider the limit distribution for a sequence of random variables\[ \alpha (\mu - n\theta ),\quad \alpha = \alpha (n),\quad \theta = \theta (n). \]
Theorems 1–5 give characteristic functions for some possible limit distributions.
The main result of this paper is Theorem 6:
If the limit distribution for$\alpha (\mu - n\theta )$exists, then it does not differ from one of those ound in Theorems 1–5 by more than a linear transformation.

Continuous Generalizations of Chebyshev’s Inequality

P. Whittle

Theory Probab. Appl. 3, pp. 358-366 (9 pages) | Cited 3 times

Online Publication Date: July 17, 2006

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Let $x(t)$ be a random function with known ${\bf E}[x(t)]$ and ${\bf E}[x(t)x(s)]$, $0 \leqq s$, $t \leqq 1$. In Section 3 a bound is given for the probability that $|x(t)|$ exceeds the given function $\alpha (t)$ at least for one $t$. The bound involves an arbitrary quadratic form, which can be selected in an appropriate way giving certain bounds (see, for example, formula (30)). The effectiveness of this method depends on the degree of differentiability of $x(t)$. In the last two sections the case is treated when $x$ is a function of several variables $t_1 ,t_2 , \cdots ,t_m $.

Information Transmission in a Channel with Feedback

R. L. Dobrushin

Theory Probab. Appl. 3, pp. 367-383 (17 pages) | Cited 5 times

Online Publication Date: July 17, 2006

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In this paper we prove that the use of feedback does not increase the capacity of channels without memory. We also consider some simple channels with memory and compare their capacities when feedback is and is not used.

On the Distribution of Sums of Random Variables Defined on a Homogeneous Markov Chain with a Finite Number of States

I. S. Volkov

Theory Probab. Appl. 3, pp. 384-399 (16 pages) | Cited 6 times

Online Publication Date: July 17, 2006

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Local and integral limit theorems are established for a non-periodical case. The results are given in the form of asymptotic expansions taking into account various possible values of the sums under consideration.

Diffusion Processes and Elliptic Differential Equations Degenerating at the Boundary of the Domain

R. Z. Khas’minskii

Theory Probab. Appl. 3, pp. 400-419 (20 pages) | Cited 9 times

Online Publication Date: July 17, 2006

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In this paper Markov diffusion processes with continuous paths are studied. We give the definitions of attracting, repelling, unattainable and regular boundaries.
Effective sufficient conditions for each type expressed in terms of the coefficients of the equation (2) are given also. These conditions are also necessary for additional assumptions.

Final Probabilities for Multi-Dimensional Markov Processes Which Describe the Action of Some Two-Stage Telephone Systems with Busy-Signals

G. P. Basharin

Theory Probab. Appl. 3, pp. 420-425 (6 pages) | Cited 1 time

Online Publication Date: July 17, 2006

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Final probabilities are determined for mufti-dimensional Markov processes with continuous time and a finite number of states describing the action of a two-stage telephone system with one switch in the second stage.
It is assumed that service calls form independent Poisson streams of calls and that the service times have independent negative exponential distributions.
On the basis of these final probabilities some other probability formulas for a common number of the busy lines are determined. These formulas are extensions of the well known Erlang’s formulas in the mufti-dimensional case. Common group selection and random occupation of each free connecting device are considered.

On Multi-Dimensional Stationary Random Processes

E. G. Gladyshev

Theory Probab. Appl. 3, pp. 425-428 (4 pages)

Online Publication Date: July 17, 2006

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The necessary and sufficient conditions for the regularity as well as for “the maximum regularity” of $n$-dimensional stationary random processes $(x_1 (t),x_2 (t), \cdots ,x_n (t))$ with continuous time are obtained.
Wold’s development “for the process” is also proved.

A Simplified Method of Experimentally Evaluating the Entropy of a Stationary Sequence

R. L. Dobrushin

Theory Probab. Appl. 3, pp. 428-430 (3 pages) | Cited 1 time

Online Publication Date: July 17, 2006

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In this paper we relate a new method for evaluating the entropy of a stationary sequence using a given sample. This method is simpler than the one usually used.

Sequential Selection between Two Solutions for a Poisson Process

V. S. Mikhalevich

Theory Probab. Appl. 3, pp. 430-434 (5 pages) | Cited 1 time

Online Publication Date: July 17, 2006

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In this paper the results of Theorem 4.1 in [1] are extended to arbitrary a priori distributions.

On Uniform Approximation of the Binomial Distribution by Infinitely Divisible Laws

I. P. Tsaregradskii

Theory Probab. Appl. 3, pp. 434-438 (5 pages) | Cited 9 times

Online Publication Date: July 17, 2006

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Let $F_p^n (x)$ be an $(n,p)$–binomial distribution function and be the set of all infinitely divisible laws. We define\[ \rho \left( {F_p^n ,\mathfrak{G}} \right) = \mathop {\inf }\limits_{G \in \mathfrak{G}} \mathop {\sup }\limits_x \left| {F_p^n (x) - G(x)} \right|. \]
Then,\[ \mathop {\sup }\limits_{0 \leqq p \leqq 1} \rho \left( {F_p^n ,\mathfrak{G}} \right) < \frac{{C_0 }} {{\sqrt n }}, \]where $C_0 $ is an absolute constant.

Reviews and Bibliography

Theory Probab. Appl. 3, pp. 438-440 (3 pages)

Online Publication Date: July 17, 2006

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